A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem

Benamou, Jean-David; Brenier, Yann

doi:10.1007/s002110050002

Cited by 1,373 publications

(1,673 citation statements)

References 31 publications

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“…Monge-Kantorovich problem has been studied in statistics, functional analysis, and the atmospheric sciences; see [11], [5] and the references therein. A fundamental theoretical result [22], [6], [15] …”

Section: £ ¡mentioning

confidence: 99%

Optimal Mass Transport for Registration and Warping

Haker

Zhu

Tannenbaum

et al. 2004

International Journal of Computer Vision

352

356

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Image registration is the process of establishing a common geometric reference frame between two or more image data sets possibly taken at different times. In this paper we present a method for computing elastic registration and warping maps based on the Monge-Kantorovich theory of optimal mass transport. This mass transport method has a number of important characteristics. First, it is parameter free. Moreover, it utilizes all of the grayscale data in both images, places the two images on equal footing and is symmetrical: the optimal mapping from image to image ¡ being the inverse of the optimal mapping from ¡ to £ ¢ The method does not require that landmarks be specified, and the minimizer of the distance functional involved is unique; there are no other local minimizers. Finally, optimal transport naturally takes into account changes in density that result from changes in area or volume. Although the optimal transport method is certainly not appropriate for all registration and warping problems, this mass preservation property makes the Monge-Kantorovich approach quite useful for an interesting class of warping problems, as we show in this paper. Our method for finding the registration mapping is based on a partial differential equation approach to the minimization of the ¤ ¦ ¥ Kantorovich-Wasserstein or "Earth Mover's Distance" under a mass preservation constraint. We show how this approach leads to practical algorithms, and demonstrate our method with a number of examples, including those from the medical field. We also extended this method to take into account changes in intensity, and show that it is well suited for applications such as image morphing.

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Section: £ ¡mentioning

confidence: 99%

Optimal Mass Transport for Registration and Warping

Haker

Zhu

Tannenbaum

et al. 2004

International Journal of Computer Vision

352

356

View full text Add to dashboard Cite

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“…Consequently, by Lemma A.2 γ Ω Ω = (S, Id) # ν + τ h, for a map S that is optimal in the classical sense and is uniquely defined a.e. Thus, it follows from (3.19) and (3.20) that |y| 2 2 − τ ϕ * (y) and τ e ′ • h(y) + |y| 2 2 are Lipschitz, and have the same derivative a.e in Ω. Therefore, we deduce that there exists a constant κ such that…”

Section: Properties Of W B Eψτmentioning

confidence: 80%

A new family of transportation costs with applications to reaction–diffusion and parabolic equations with boundary conditions

Morales

2018

Journal de Mathématiques Pures et Appliquées

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Abstract. This paper introduces a family of transportation costs between non-negative measures. This family is used to obtain parabolic and reaction-diffusion equations with drift, subject to Dirichlet boundary condition, as the gradient flow of the entropy functional Ω ρ log ρ + V ρ + 1 dx. In [5], Figalli and Gigli study a transportation cost that can be used to obtain parabolic equations with drift subject to Dirichlet boundary condition. However, the drift and the boundary condition are coupled in that work. The costs in this paper allow the drift and the boundary condition to be detached.

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“…Gabriel et al (2010) and González-Hernández et al (2006) proposed a general approximation scheme for the MK problem on compact and Polish spaces, respectively. Other schemes for the MK problem were studied by Benamou and Brenier (2000), Caffarelli et al (2002), Benamou (2003), Guittet (2003), and recently by Bosc (2010) and Mèrigot (2011). This work presents an improvement to the approximation scheme for the MK problem on compact spaces given by Gabriel et al (2010).…”

Section: ML Avendaño-garrido Et Almentioning

confidence: 87%

A metaheuristic for a numerical approximation to the mass transfer problem

Avendaño-Garrido

Gabriel-Arguelles

Quintana-Torres

et al. 2016

International Journal of Applied Mathematics and Computer Science

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This work presents an improvement of the approximation scheme for the Monge-Kantorovich (MK) mass transfer problem on compact spaces, which is studied by Gabriel et al. (2010), whose scheme discretizes the MK problem, reduced to solve a sequence of finite transport problems. The improvement presented in this work uses a metaheuristic algorithm inspired by scatter search in order to reduce the dimensionality of each transport problem. The new scheme solves a sequence of linear programming problems similar to the transport ones but with a lower dimension. The proposed metaheuristic is supported by a convergence theorem. Finally, examples with an exact solution are used to illustrate the performance of our proposal.

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A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem

Abstract: The L 2 Monge-Kantorovich mass transfer problem [31] is reset in a fluid mechanics framework and numerically solved by an augmented Lagrangian method.

Cited by 1,373 publications

References 31 publications

Optimal Mass Transport for Registration and Warping

Optimal Mass Transport for Registration and Warping

A new family of transportation costs with applications to reaction–diffusion and parabolic equations with boundary conditions

A metaheuristic for a numerical approximation to the mass transfer problem

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